Tilburg University
Certification and Minimum Quality Standards when Some Consumers are Uninformed
Buehler, B.; Schütt, Florian
Publication date:
2012
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Citation for published version (APA):
Buehler, B., & Schuett, F. (2012). Certification and Minimum Quality Standards when Some Consumers are
Uninformed. (TILEC Discussion Paper; Vol. 2012-040). Tilburg: TILEC.
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TILEC
TILEC Discussion Paper
DP 2012-040
Certification and minimum quality
standards when some consumers are
uninformed
By
Benno Buehler, Florian Schuett
November 21, 2012
ISSN 1572-4042
ISSN 2213-9419 http://ssrn.com/abstract=2179028
Certification and minimum quality standards
when some consumers are uninformed∗
Benno Buehler†
Florian Schuett‡
November 2012
Abstract
We compare certification to a minimum quality standard (MQS) policy in a duopolistic
industry where firms incur quality-dependent fixed costs and only a fraction of consumers
observes the quality of the offered goods. Compared to the unregulated outcome, both
profits and social welfare would increase if firms could commit to producing a higher
quality. An MQS restricts the firms’ quality choice and leads to less differentiated goods.
This fuels competition and may therefore deter entry. A certification policy, which awards
firms with a certificate if the quality of their products exceeds some threshold, does not
restrict the firms’ quality choice. In contrast to an MQS, certification may lead to more
differentiated goods and higher profits. We find that firms are willing to comply with an
ambitious certification standard if the share of informed consumers is small. In that case,
certification is more effective from a welfare perspective than a minimum quality standard
because it is less detrimental to entry.
Keywords: Certification, minimum quality standard, unobservable quality, policy intervention
JEL classification: L15; L13; L51; D82
∗
We would like to thank Markus Reisinger, Patrick Rey, Klaus Schmidt and particularly Yossi Spiegel for
helpful comments and discussions as well as Fabian Bergès, Giacomo Calzolari, Stéphane Caprice, Joe Farrell,
Anne Layne-Farrar, Jérôme Mathis and Roland Strausz for useful suggestions. We are also grateful to seminar
participants at Tilburg University and the University of Bologna, as well as conference participants at the
EARIE congress in Stockholm and CRESSE in Chania for their comments. Buehler acknowledges financial
support from the German Science Foundation (DFG) through SFB TR/15 and from the German Academic
Exchange Service (DAAD). Schuett acknowledges financial support from the 7th European Community Framework Programme through a Marie Curie Intra European Fellowship (grant number PIEF-GA-2010-275032).
†
Department of Economics, Ludwig Maximilian University Munich, Amalienstrasse 17, 80333 Munich,
Germany and Toulouse School of Economics (IDEI), 21 allée de Brienne, 31000 Toulouse, France. E-mail:
[email protected].
‡
Tilburg University (TILEC and CentER). Postal address: Tilburg University, Department of Economics,
PO Box 90153, 5000 LE Tilburg, The Netherlands. Email: [email protected].
1
1
Introduction
When consumers are ill-informed about the quality of a product, firms may provide less than
the socially optimal level of quality.1 Policy makers have resorted to a variety of instruments
to address this problem. Two important such instruments are minimum quality standards
and certification. Minimum quality standards (MQS) prevent firms from selling goods whose
quality is below a predefined standard. Examples include safety standards as well as occupational licensing for professional services. Certification is a process whereby the government or
an independent third party verifies if a product fulfills certain criteria. Products that satisfy
these criteria obtain a certificate that is visible to consumers. Certification is used to document, among other things, the security of cars (crash tests, rollover ratings) and the origin of
food and wood (organic food certificates, FSC label for timber from sustainable forestry).2
It is natural to ask which of these instruments is likely to be most effective in a given
context. To investigate this question, we introduce uninformed consumers into the wellknown model of Ronnen (1991). We show that certification can outperform minimum quality
standards, in the sense that it induces firms to produce qualities that are closer to the socially
optimal levels and thus leads to greater welfare. The superior performance of certification
arises precisely when the underlying problem these instruments are intended to solve is most
severe, namely, when the share of uninformed consumers is large. By contrast, when the
share of uninformed consumers is small, the rationale for certification disappears. Minimum
quality standards, however, may continue to play a useful role, as Ronnen (1991) established
for the limiting case in which all consumers are informed.
Ronnen (1991) studies a duopoly model in which firms play a two-stage game. In stage
one, firms decide whether to enter the market and invest in the quality of their product. In
stage two, they observe each other’s entry and quality choices, and then compete in prices.
All consumers observe quality; they have unit demand and differ in their taste for quality.
The equilibrium is one of vertical differentiation, with one firm selling high quality at a high
price and the other firm selling low quality at a low price. We depart from Ronnen’s setup
1
Firms respond to this problem in various ways, such as by promising warranties, signaling quality through
prices, or developing a reputation for high quality. Research on these market instruments has shown, however,
that they may alleviate the problem but rarely eliminate it. See, e.g., Gal-Or (1989) for warranties and
Daughety and Reinganum (2008a,b) for price signaling.
2
A third instrument that is important in practice is mandatory disclosure, whereby firms are obliged to
publish standardized and comparable information about their products. The distinction between mandatory
disclosure and certification is not always clear-cut. There are two main features that distinguish the two: first,
disclosure usually concerns technical information which consumers may not always be able to understand and
translate into an assessment of quality, whereas certification usually involves a grading system that is easy
to grasp; second, disclosure can in principle give consumers an idea of a product’s precise position on the
quality scale, whereas certification is coarse (in the sense of dividing the scale into discrete steps) and thus
merely allows consumers to identify the interval to which a product belongs. The coarseness of certification is
important for the results in this paper.
2
by introducing a subset of consumers who do not observe quality.
In this setup, absent government intervention, firms’ revenues from uninformed consumers
do not directly depend on quality: by assumption, uninformed consumers cannot react to
changes in quality. When deciding on their investment in quality, firms thus only take into
account the revenues from informed consumers. Nevertheless, the uninformed consumers
correctly anticipate the firms’ quality choices, so that in equilibrium the firms’ quality investments determine their revenue from both types of consumers. The firms would like to commit
to producing higher quality, but have no credible way of doing so. It follows that quality
levels are below those that would be optimal for the firms. While the equilibrium continues
to feature a vertical-differentiation outcome, both firms’ qualities are lower than under full
information. We then study the effect of the two policy instruments considered above. Our
use of the Ronnen framework, in which an MQS is known to raise welfare when all consumers
are informed, gives an MQS its best shot.3
Certification offers firms the possibility to commit to a higher level of quality. When
the certification threshold is set above the highest quality level that would be offered in an
unregulated market, it may induce both firms to raise their quality. With the certificate, a
firm can demonstrate that the quality of its product meets at least the certification standard.
Therefore, if the standard is not too demanding, one firm will match this level. The other firm
differentiates its product by offering low quality, but compared to the unregulated equilibrium,
it can raise its quality without compromising the degree of differentiation between the goods
(and thus without intensifying price competition). As we show, there exists a certification
standard that increases both consumer surplus and industry profit.
When setting the certification standard, the government has to ensure that obtaining the
certificate is attractive for producers. The certification threshold has to be set such that
the required investment in quality is not too high compared to the expected revenues. The
decision to certify a product crucially depends on the profit that a firm expects to earn from
selling its product without the certificate. As argued above, the quality level the firms can
credibly produce without the certificate (and the resulting profit they can secure) is low when
the fraction of informed consumers is small. This leads to the surprising result that if quality
is more difficult to discern, a firm is more reliant on certification and tends to comply with a
higher certification standard.
The government can exploit this informational problem. Through the strategic use of
certification, it can achieve a higher maximum quality than in case the quality is perfectly
observable.4 In our framework, this is welfare-enhancing because even full-information qual3
A priori, the welfare effect of an MQS is ambiguous; see Leland (1979) and Shapiro (1983).
Indeed, concerning the FSC certificate mentioned above, the timber industry’s attempts to install another
certification system with a relatively soft standard provide evidence that some firms would prefer less restrictive
4
3
ities are lower than the socially optimal ones. The source of this underprovision of quality is
twofold. It stems both from the fact that the marginal consumer has a lower taste for quality
than the average consumer (see also Spence, 1975), and from the firms’ desire to offer differentiated products so as to alleviate price competition. Importantly, our results also apply
to other sources of quality underprovision. For example, if underprovision occurs because of
positive externalities (e.g., car safety), the government could similarly raise the quality level
by resorting to certification if consumers are not fully informed.
The impact of an MQS differs from that of certification. By restricting the admitted
quality range and thus the firms’ ability to differentiate their goods, an MQS intensifies
price competition and requires higher minimum investments in quality. Therefore, adopting
an MQS reduces the firms’ profits and may deter them from entering. We show that this
problem is particularly severe when quality is difficult to observe, so that the firms’ ability to
credibly produce high quality is limited. By contrast, our analysis emphasizes that suitable
certification does not restrict entry. We thus conclude that suitable certification may improve
welfare more than an MQS if only few consumers are informed. If instead almost all consumers
can observe the actual quality, firms have no need to rely on certification. The welfare gains
from adopting a suitable MQS are then higher.
Related Literature.
Our model is closely related to the literature on oligopolistic com-
petition and minimum quality standards in markets for vertically differentiated goods. The
insight that vertical product differentiation can be used by oligopolistic firms in order to relax
price competition is due to Gabszewicz and Thisse (1979) and Shaked and Sutton (1982).5
Based on this insight, Ronnen (1991) demonstrates that the government can increase welfare by adopting an MQS if improving a product’s quality requires fixed investments but no
variable costs.6 Crampes and Hollander (1995) address the same question when firms incur
certification. For a detailed report, see http://www.fern.org/sites/fern.org/files/media/documents/document
1890 1900.pdf. The observation that these attempts have been ineffective so far exemplifies that firms often
lack the power to install certification in which consumers have confidence. This in turn allows the government
– or institutions which may credibly enforce certification – to step in and manipulate the qualities in the
market by setting suitable certification standards. This may also explain why simple threshold schemes are so
popular.
Our analysis is based on the assumption that policy interventions are adopted so as to maximize social
welfare. Although we speak of governmental interventions for concreteness, our results also apply to nongovernmental institutions whose objective is to maximize social welfare. For example, the FSC is a nongovernmental, not-for-profit organization that promotes “environmentally appropriate, socially beneficial, and
economically viable management of the world’s forests” (see http://www.fsc.org/vision mission.html). This
objective could be interpreted as maximizing social welfare.
5
Wauthy (1996) endogenizes the market coverage.
6
The same setup, which was originally inspired by Tirole (1988), has also been analyzed by Choi and Shin
(1992). Lehmann-Grube (1997) demonstrates that the high quality provider usually earns higher profits and
shows that this result survives when firms choose their quality sequentially. In a slightly modified setup, Motta
(1993) compares price and quantity competition.
4
variable costs for quality and obtain the same qualitative effect of introducing an MQS if the
costs of quality are convex enough.7
Our analysis is also related to the literature on certification. Assuming perfect commitment and treating the seller’s quality as exogenously determined, Lizzeri (1999) studies the
profit maximizing policy of a monopolistic and of oligopolistic certifiers. Based on a similar
framework, Albano and Lizzeri (2001) endogenize the seller’s quality choice. Strausz (2005)
as well as Mathis, McAndrews, and Rochet (2009) concentrate on the incentives of certifiers
to honestly rate the seller’s quality. All of these models focus on the behavior of certifiers and
consider a rather simple structure of the market for the rated goods.8 In contrast, we abstract
from problems associated with dishonest certification and focus on how costless certification
affects the rated firms’ behavior in a more complex competitive environment.
The remainder of this paper is organized as follows. Section 2 sets out the model. Section 3 derives the equilibrium prices for given quality choices. Section 4 solves for the equilibrium qualities and the number of active firms in the absence of any intervention. Section 5
investigates the effects of government intervention on the equilibrium quality choices and
characterizes the welfare maximizing certification standard and minimum quality standard.
It also compares the impact of an MQS to that of certification. Finally, Section 6 concludes.
All proofs are relegated to Appendix A.
2
The model
There are two identical potential entrants to the market. Each firm i can offer a single quality
qi ∈ [0, ∞). The cost of installing the production technology to produce goods of quality q
is C(q), where C(0) = C 0 (0) = 0, C 0 (q) > 0 for q > 0, C 00 > 0, and limq→∞ C 0 (q) = ∞. For
technical reasons, we also assume C 000 ≥ 0. Once the production technology is in place, actual
production is costless (i.e., there are no variable production costs).
There is a mass one of consumers with unit demand. A consumer who buys a product of
quality q at price p has utility θq − p, where θ is the consumer’s taste for quality. We assume
that θ is uniformly distributed on [0, 1]. Consumers differ in their information about quality.
A fraction α ∈ (0, 1] of consumers, which we refer to as informed, observe the quality of the
products on offer before deciding whether and from which firm to buy. A fraction 1 − α of
consumers, which we refer to as uninformed, do not observe quality before buying.9 Since
7
Kuhn (2007) shows that an MQS may be detrimental in a setup with variable cost of quality if consumers
derive some baseline benefits from the consumption of the good that is independent from its quality. Valletti
(2000) demonstrates that a mildly restrictive MQS unambiguously reduces total welfare when firms compete
in quantities instead of prices.
8
Similar to our model, Heyes and Maxwell (2004) consider the effect of MQS and certificates, but do not
model the marked for the rated goods explicitly.
9
We assume for simplicity that whether a consumer is informed or uninformed is independent of θ.
5
all uninformed consumers have the same information, they form the same belief q̂i about
the expected quality of the good offered by each firm i. All consumers, whether informed
or uninformed, can identify by which firm the goods on offer have been produced (perhaps
because goods carry the producer’s name).
We consider two instruments of government intervention:10 certification and a minimum quality standard (MQS). Certification entails that the government publicly announces a
threshold q C . Any firm that installs a quality q ≥ q C obtains the certificate, which is observed
by all consumers and firms. The government does not disclose the exact quality of the firms
that are awarded the certificate. An MQS entails that the government chooses a threshold
q M QS below which firms cannot be active. Only firms with a quality q ≥ q M QS are allowed
to sell their product to consumers. Thus, the difference between certification and an MQS
is that under certification, firms whose quality is below the standard are still allowed to be
active, whereas with an MQS they are not.
The timing is as follows (see Figure 1). If a certification or MQS policy is in place, the
government announces the respective threshold (q C or q M QS ) before the start of the game.
At t = 0, firms simultaneously decide whether or not to enter the market. Firms that enter
also choose the quality of their production technology. If an MQS is in place, they must
choose a quality that weakly exceeds q M QS . If a certification policy is in place, all firms
that have installed at least q C obtain the certificate. Each firm in the market learns whether
its competitor has entered but does not observe the competitor’s quality. Consumers and
firms observe which firms have obtained a certificate. Because at this stage firms have the
same information about their competitor as the uninformed consumers, we assume that they
form the same belief about quality. At t = 1, uninformed consumers enter the market. The
active firms simultaneously set first-period prices. Based on the prices and on their beliefs,
uninformed consumers decide whether and from which seller to buy. At t = 2, informed
consumers enter the market. Each firm learns its competitor’s quality. Firms simultaneously
set second-period prices, and the informed consumers decide whether and from which seller
to buy. For simplicity, firms do not discount their profits.
The sequential structure of the game captures the idea that early consumers of an experience good have less information about its quality than late consumers, who can learn
from the experience of early consumers (e.g., through word of mouth).11 We assume that
all consumers are short lived and have to make their decision in the period they enter the
10
We use the term government intervention in order to highlight that these instruments are designed to
increase social welfare. We abstract from problems such as commitment and honesty, among other things.
11
The assumption that firms can change prices but not qualities between periods 1 and 2 can be justified by
the fact that price changes can be done quickly while a change of the production technology is usually more
time-consuming.
6
Firms decide on entry,
observe entry decisions,
invest in quality.
Firms set period-1 prices.
Uninformed consumers make
purchasing decisions.
Firms observe qualities,
set period-2 prices.
Informed consumers make
purchasing decisions.
t=0
t=1
t=2
time
Figure 1: Timing of the game
market, which is exogenously determined.
If a single firm enters, we use the index M . In case two firms enter, we assign the firm
that produces a weakly higher quality the index H and the remaining one the index L so
that qH ≥ qL . Accordingly, we will refer to firm H as the high-quality producer and to
firm L as the low-quality producer. The equilibrium concept we use is Perfect Bayesian
Equilibrium. We focus on pure strategy equilibria and impose the refinement that beliefs are
passive: consumers and competitors do not revise their beliefs about a firm’s quality when it
charges an unexpected price. We discuss this assumption in detail in the following section.12
In equilibrium, the beliefs (q̂L , q̂H ) or q̂M have to be correct, consumers make the utility
maximizing choice, and each firm’s strategy maximizes its profits given the beliefs and the
equilibrium strategies of the competitor and the consumers.
3
Price Equilibrium
We first examine the equilibrium prices that obtain in period two, when only informed consumers are in the market. The second-period pricing subgame is equivalent to the setup in
Ronnen (1991) (where all consumers know the quality of the offered products). Let x ≡ p/q
denote the quality-deflated or hedonic price. A consumer with taste for quality θ gets positive
utility from buying if and only if θq − p ≥ 0 ⇔ θ ≥ x. Thus, if a single firm has entered, the
optimal quality-deflated price is x∗M = 1/2. If both firms have entered, let r ≡ qH /qL denote
the relative quality of their products. If qH > qL , a consumer with taste θ prefers H to L if
and only if θqH − pH ≥ θqL − pL or, equivalently, θ ≥
pH −pL
qH −qL
≡ θ̂. Ronnen (1991) provides a
detailed derivation of the following result:
Lemma 1 (Ronnen (1991)). Suppose that both firms have entered the market with qH ≥ qL
and that consumers observe qualities. In the unique price equilibrium, the quality-deflated
prices of firm L and H are x∗L =
at θ̂∗ =
r−1
4r−1
and x∗H =
2(r−1)
4r−1 .
The indifferent consumer is located
Consumers with θ ∈ [0, x∗L ) do not buy, those with θ ∈ [x∗L , θ̂∗ ) buy from firm
h
i
L, and those with θ ∈ θ̂∗ , 1 buy from firm H.
12
2r−1
4r−1 .
Passive beliefs are common in the literature on vertical contracts, as discussed in Rey and Tirole (2007).
7
Thus, for r > 1, the equilibrium of the pricing subgame at t = 2 is such that the consumers
with the lowest taste for quality abstain from buying, the consumers with the highest taste
for quality buy the high-quality good, while those with intermediate taste for quality buy the
low-quality good. Formally, the equilibrium entails x∗L < x∗H < θ̂∗ .
From Lemma 1 we can derive the firms’ equilibrium revenues per consumer as
r(r − 1)
RL (qL , qH ) ≡ qL x∗L θ̂∗ − x∗L = qL
(4r − 1)2
4r(r − 1)
RH (qL , qH ) ≡ qH x∗H 1 − θ̂∗ = qH
.
(4r − 1)2
(1)
(2)
Total revenues from period two are obtained by multiplying these expressions by the share
of informed consumers, α. A firm’s revenue increases when holding its own quality fixed and
raising the degree of disparity r. The reason is that more differentiated products give rise to
higher equilibrium prices. If both firms produce the same quality, then the equilibrium prices
are zero. Letting subscripts denote partial derivatives, straightforward calculations show that
revenues satisfy the following properties, for i = L, H:13
Rqi i > 0 > Rqi i ,qi
Rqi i ,qj > 0,
i 6= j.
That is, a firm’s revenue is increasing and concave in its own quality, and a firm’s marginal
revenue is increasing in its competitor’s quality (qualities are strategic complements). The
intuition for strategic complementarity is the following. As the low-quality firm raises qL ,
the high-quality firm has a stronger incentive to raise qH to escape price competition. As
the high-quality firm raises qH , price competition is alleviated, so the low-quality firm has a
stronger incentive to raise qL to capture a larger share of demand.
We now turn to the equilibrium prices that obtain in period one, when only uninformed
consumers are in the market. Note first that period-one prices cannot signal quality. Our setup
deliberately rules out all channels through which signaling could be sustained: marginal costs
do not depend on quality, as in Daughety and Reinganum (2008b) or Janssen and Roy (2010);
firms do not observe each others’ quality before setting period-one prices, as in Hertzendorf
and Overgaard (2001) or Fluet and Garella (2002); informed consumers are not in the market simultaneously with uninformed consumers, as in Yehezkel (2008). Ruling out signaling
considerably simplifies the analysis. It can also be justified on the grounds that we are most
worried about those markets where distortions caused by asymmetric information cannot be
13
2
The marginal revenues of the low-quality and high-quality firm with respect to own quality are RqLL =
r (4r−7)
(4r−1)3
2
)
and RqHH = 4r(2−3r+4r
. They depend on qL and qH only through the relative quality r. Formally,
(4r−1)3
marginal revenues are homogeneous of degree zero.
8
alleviated through signaling: when signaling is possible, the market may be able to solve
(or at least mitigate) the underlying information problem without the need for government
intervention.
The absence of signaling is why the restriction to equilibria with passive beliefs is reasonable. As is well known, the PBE concept suffers from the problem of multiple equilibria.
In our setup, the multiplicity problem is particularly severe: without restrictions on outof-equilibrium beliefs, (almost) any period-one prices can be supported as an equilibrium.14
Given that prices cannot signal qualities, however, the most plausible outcome is one in which
consumers’ beliefs do not depend on period-one prices. Consumers know that prices do not
convey any information about quality. Thus, it can be argued that they should not revise
their beliefs after observing an unexpected price. This is precisely what occurs under our
passive-beliefs assumption.15 The equilibrium in which beliefs about qualities do not depend
on period-one prices is also the only one that satisfies In and Wright’s (2011) concept of
reordering invariance.16
Note that the restriction to passive beliefs does not affect the equilibrium qualities. Since
prices do not signal qualities, the actual quality only affects the firms’ revenues from informed
consumers at t = 2. Revenues from uninformed consumers depend on beliefs about quality,
but not on quality itself.17 We analyze the choice of quality in the following section.
Since consumers do not revise their belief q̂i after observing unexpected prices, the equilibrium prices x∗i at t = 1 are determined by the formulas given by Lemma 1 after replacing
qi by q̂i . Similarly, revenues per consumer are given by expressions (1) and (2), replacing qi
by q̂i . Firm i’s total revenue for beliefs (q̂L , q̂H ) and actual qualities (qL , qH ) is
(1 − α) Ri (q̂L , q̂H ) + αRi (qL , qH ), i ∈ {L, H} .
14
It suffices that consumers believe that any firm that deviates has a quality of zero.
Because of our assumption that consumers can identify the producer of a good, their beliefs about the
firms’ qualities may differ even though the firms are ex-ante identical. In particular, our restriction to passive
beliefs implies that if firms produce vertically differentiated goods in equilibrium but the low-quality firm
deviates and charges the same price as its competitor, the consumers maintain their beliefs about this firm.
16
Our setup calls for the firms to make two sequential moves that are not observable to either consumers or
competitors (choosing qualities and then choosing period-one prices). Simply put, reordering invariance selects
the equilibria that survive a reordering of these moves (i.e., firms choosing period-one prices before choosing
qualities). In our game, reordering creates proper subgames (one corresponding to each period-one price), so
that subgame perfection can be applied. Since the equilibrium strategy prescribes the same quality whatever
the period-one price (see Section 4), a belief that assigns positive probability to any other quality after a
deviation is not derived from equilibrium strategies, and can therefore not be part of a reordering-invariant
PBE.
17
The beliefs do affect the equilibrium prices and thus the profits in the second stage, which has an impact
on the firms’ decision to enter the market and on welfare.
15
9
4
Quality choice and entry without intervention
This section examines the equilibrium quality levels in the absence of policy intervention. We
first derive the quality choice that maximizes firm L’s profit given firm H’s quality, qH , and
beliefs (q̂L , q̂H ), which solves
max (1 − α) RL (q̂L , q̂H ) + αRL (qL , qH ) − C(qL ).
qL
(3)
Let bL (qH , α) denote firm L’s restricted best response to quality qH given α. That is, bL is
the quality qL that solves (3) subject to qL ≤ qH . The first-order condition
αRqLL (qL , qH ) = C 0 (qL )
(4)
uniquely characterizes the best response of firm L whenever bL (qH , α) < qH .
The quality choice that maximizes firm H’s profit given qL and beliefs (q̂L , q̂H ) solves
max (1 − α)RH (q̂L , q̂H ) + αRH (qL , qH ) − C(qH ).
qH
(5)
Let bH (qL , α) denote firm H’s restricted best response to quality qL given α, which is the
quality qH that solves (5) subject to qH ≥ qL . The first-order condition
αRqHH (q̂L , qH ) = C 0 (qH )
(6)
uniquely defines the best response of firm H whenever bH (qL , α) > qL .18
In equilibrium, uninformed consumers correctly anticipate firms’ quality choices. Letting
Πi (q
L , qH )
denote the profits of firm i when beliefs are correct, we thus have
Πi (qL , qH ) = (1 − α)Ri (qL , qH ) + αRi (qL , qH ) − C(qi ) = Ri (qL , qH ) − C(qi ).
The following Proposition extends Theorem 1 in Ronnen (1991) to the case where 0 < α < 1.
Proposition 1. For any α ∈ (0, 1], there is a unique equilibrium in quality choices, charac∗ ) that solve conditions (4) and (6). When consumers correctly anterized by qualities (qL∗ , qH
∗ ) > ΠL (q ∗ , q ∗ ) > 0.
ticipate that the firms produce at these quality levels, profits are ΠH (qL∗ , qH
L H
∗ increase in the fraction of informed consumers α.
The quality levels qL∗ and qH
Proposition 1 states that the quality-choice game without government intervention has
a unique equilibrium that is characterized by vertical differentiation: firm H best-responds
to qL by offering a quality above qL , and firm L best-responds to qH by offering a quality
below qH . This result is illustrated in Figure 2. The proposition also shows that the highquality firm earns higher profits than the low-quality firm and that both earn positive profits.
18
In the proof of Proposition 1 we show that bH (q̂L , α) always exists and is bounded above.
10
qH
bL (qH , α)
bH (qL , α0 )
bL (qH , α0 )
bH (qL , α)
∗
∗
(qL
(α0 ), qH
(α0 ))
∗
∗
(qL
(α), qH
(α))
45◦
qL
Figure 2: Equilibrium in quality choices: effect of an increase in α (α0 > α)
To see that the low-quality firm’s profit is positive, rewrite the equilibrium profit as ΠL∗ =
R qL∗ L
∗ ) − C 0 (q) dq. The optimality condition (4) implies that the marginal profit
RqL (q, qH
0
∗ ). The profit of
with respect to qL , RqLL (qL , qH ) − C 0 (qL ), must be non-negative at (qL∗ , qH
firm L is thus positive, because each firm’s marginal profit decreases in the own quality. Since
both firms earn positive profits, we may conclude that it is optimal for them to enter the
market. The proof of Proposition 1 also establishes that firm H has no incentive to deviate to
a quality level below its competitor’s. The assumption C 000 ≥ 0 ensures that the low-quality
firm has no incentive to deviate to a quality level above its competitor’s.
The result that equilibrium qualities increase with α can be explained as follows. Since
marginal revenue is downward sloping in the own quality level, conditions (4) and (6) imply
that the restricted best responses increase in the fraction of informed consumers when holding
H
the competitor’s quality constant. Formally, bL
α (qH , α) ≥ 0 and bα (qL , α) ≥ 0 with a strict
inequality if bL (qH , α) < qH and bH (qL , α) > qL , respectively. Intuitively, an increase in α
raises the marginal benefit of producing higher quality because a larger fraction of consumers
can react to the increase. This effect is reinforced by strategic complementarity: since Rqi i ,qj >
0, each firm has an incentive to further raise its quality following an increase in its competitor’s
quality.
Importantly, when α < 1, each firm could improve its situation if it were able to commit to producing higher quality: in equilibrium, the beliefs are correct, and the optimality
∗ ) = (1 − α) Ri (q ∗ , q ∗ ) > 0. Thus, if a firm inconditions (4) and (6) imply that Πiqi (qL∗ , qH
qi L H
creased its quality slightly and consumers adapted their beliefs accordingly, the firm’s profit
11
would increase. However, the proportion 1 − α of consumers does not react to changes of
the actual quality of a good, so that firms have insufficient incentives to invest in quality.
Each firm installs a production technology that maximizes αRi (qL , qH ) − C(qi ) instead of the
whole profit Ri (qL , qH ) − C(qi ). In particular, if consumers expected a firm to produce at the
profit-maximizing quality level, then this firm would have an incentive to deviate to a lower
quality level.19
∗ so as to maximize
For later reference, we note that a monopolist sets the quality level qM
αRM (qM ) − C(qM ). From limqL →0 RH (qL , qM ) = RM (qM ) and RqHH ,qL > 0, it follows that
for any α > 0, the equilibrium quality level of the high-quality firm exceeds the equilibrium
quality level of a monopolist.
In order to assess the scope for government intervention, we now relate the equilibrium
qualities to those that maximize social welfare. We define welfare as the difference between
the aggregate value of consumption and the cost of supply. Suppose two firms have entered
the market. We have
Z
W ≡ qL
θ̂∗
x∗L
Z
θdθ + qH
1
θ̂∗
θdθ − C(qL ) − C(qH ).
For any pair of qualities (qL , qH ) that are correctly anticipated by consumers, social welfare
at the ensuing equilibrium prices is
qH 12r2 − r − 2
− C(qL ) − C(qH ).
W (qL , qH ) =
2(4r − 1)2
(7)
Similarly, revenue and welfare under monopoly are RM (qM ) ≡ qM x∗M (1 − x∗M ) = 14 qM and
W M (qM ) = 83 qM − C(qM ), respectively.20
We will refer to a pair of quality levels as second-best if they solve
max W (qL , qH ).
qL ,qH
That is, the second-best qualities are those that a social planner who controls qualities but
not prices would choose.21 It is straightforward to verify that the partial derivatives of the
H 22
welfare function in (7) exceed the respective marginal profits: WqL > ΠL
qL and WqH > ΠqH .
∗ ) = (1 − α)Ri (q ∗ , q ∗ ) ≥ 0
The firms’ first order conditions (4) and (6) imply that Πiqi (qL∗ , qH
qi L H
for i ∈ {L, H}. Thus, locally increasing each firm’s quality raises welfare. In addition, as we
show in the proof of the following Lemma, W (qL , qH ) is strictly concave and WqL ,qH > 0. It
follows that the second-best qualities exceed the equilibrium qualities.
19
A similar result has been discussed by Shapiro (1982) in a monopoly setup.
These formulae obtain from the ones already presented in the limit for qL → 0.
21
Clearly, if the planner could also determine the firms’ prices, it would be optimalR that a single firm serves
0
1
the whole market at a price of zero. The first best quality level satisfies C (q F B ) = 0 θdθ = 12 .
22
See the proof of Lemma 2.
20
12
Lemma 2. For all α ∈ (0, 1], both firms’ equilibrium qualities are lower than the second-best
∗ < q SB .
qualities: qL∗ < qLSB and qH
H
According to Lemma 2, the equilibrium qualities are always too low from a social point
of view. This is true even when consumers are fully informed (α = 1). The reason for this
result is that in our setup the social incentive to raise quality exceeds the private incentive.
Appendix B shows that the difference between the marginal effect of an increase in qH on
welfare and profit can be decomposed into two parts. The first corresponds to the familiar
divergence in tastes for quality between the average and marginal consumer, which gives the
social planner a stronger incentive to increase quality than the firm. The second is the effect
on equilibrium prices. The firm has an incentive to raise quality to soften price competition,
which is absent from the planner’s considerations. It turns out that in our setup the first effect
dominates the second (see the proof of Lemma 2). The presence of uninformed consumers
reinforces this effect. While the private incentive to raise quality decreases as the share of
informed consumers shrinks, the social incentive remains unchanged. As a result, the problem
of underprovision of quality – though present under full information – is most severe when
the share of informed consumers is small.
5
The effect of government intervention
5.1
Certification
Under certification, the government announces a certification threshold q C and awards a
quality certificate to all firms whose quality exceeds q C , without revealing further information
about the firms’ qualities. Let si ∈ {0, 1} denote the outcome of certification for firm i, where
si = 1 means a certificate is awarded and si = 0 means no certificate is awarded to firm i.
Consumers incorporate the additional information made available by the certification system. Since we assume that the government does not make certification mistakes, it is natural
to impose that certification restricts out-of-equilibrium beliefs: when a firm deviates and
unexpectedly obtains (or unexpectedly does not obtain) a certificate, beliefs should be compatible with the certification outcome. Formally, let q̂i (si ) denote the belief about the quality
of firm i contingent on the certification outcome si . Beliefs are in line with the certification
outcome if q̂i (1) ∈ [q C , ∞) and q̂i (0) ∈ [0, q C ) for i ∈ {L, H}.
In order to avoid implausible equilibria, we restrict attention to equilibria that survive the
following natural refinement of out-of-equilibrium beliefs:23 after an unexpected certification
23
For example, the out-of equilibrium belief q̂H (0) = 0 implies a harsh punishment in case firm H is unexpectedly not awarded the certificate. As an implausible consequence, a firm would meet almost any certification
threshold as long as it earns non-negative profits to avoid the “stigma” of not getting the certificate.
13
outcome, consumers believe that firms best-respond to the competitor’s quality from the set
of qualities compatible with the certification outcome. Thus, if a firm unexpectedly does not
obtain the certificate, the belief has to coincide with the profit-maximizing quality within
the interval [0, q C ), holding fixed the belief about the rival’s quality.24 Likewise, if a firm
unexpectedly obtains the certificate, consumers believe that it produces the profit-maximizing
quality level within the interval [q C , ∞).25
In any equilibrium, a firm either produces a quality that satisfies the first-order condition
αRqi i (qL , qH )
0
= C (qi ) or one that exactly matches q C . To see that firms produce at no other
quality level in equilibrium, suppose that consumers (and the competitor) believed that firm
i offers a different quality. Then, it would be profitable for firm i to deviate to some quality
close to the anticipated level without changing the certification outcome and thus without
changing the uninformed consumers’ beliefs. Moreover, there is no equilibrium in which both
firms produce goods of quality q C since Bertrand competition would drive prices down to
zero so that the firms would make a loss. Therefore, at most one firm will exactly match the
certification threshold.
We restrict attention to a scenario where the government targets the high-quality firm by
setting a certification standard which is above firm H’s equilibrium quality absent interven∗ . The rationale for this approach is that when the government targets the
tion, i.e., q C ≥ qH
low-quality firm, an MQS is more efficient than certification.
In a certification equilibrium, both firms enter the market, the high-quality firm exactly
matches the certification standard, and the low-quality firm best-responds. Formally, the
ce = q C and q ce = bL (q C , α), where the superscript ce indicates
firms produce the qualities qH
L
the presence of a certification mechanism. Since we focus on certification standards that
exceed the unregulated highest quality in the market, it is intuitive that the high quality firm
will not produce a quality level that is strictly above q C . For neither firm to want to deviate,
the following conditions must hold:
ΠL qLce , q C ≥ ΠH q C , bH (q C , α)
ΠH qLce , q C ≥ max ΠH qLce , bH (qLce , α) , ΠL bL (qLce , α), qLce
24
(8)
(9)
While the profit-maximizing quality within the interval [0, q C ) may not exist for low values of q C , this
theoretical problem does not arise for certification levels that are designed to increase welfare, and can thus
be safely ignored.
25
H
Define ϕ(q1 , q2 ) ≡ αRL (q1 , q2 ) −C(q1 ) if q1 ≤ q2 and ϕ(q
) − C(q1 ) if q1 > q2 . Formally,
1 , q2 ) ≡ αR (q2 , q1
our refinement requires q̂i (0) = min arg maxq∈[0,qC ) ϕ(q, qj∗ ) and q̂i (1) = min arg maxq∈[qC ;∞) ϕ(q, qj∗ ) for
i, j ∈ {L, H}, j 6= i. In our setup this refinement is easy to apply, since the best response of firm i does
not depend on the others’ belief about this firm’s quality q̂i . Moreover, it is appealing because the revised
beliefs after an unexpected certification outcome about the deviant’s quality coincide with the most profitable
deviation. Thus, this refinement leads to “more consistency” off the equilibrium path, as is the case with In
and Wright’s 2011 concept of reordering invariance.
14
Condition (8) ensures that the low-quality firm has no incentives to “leapfrog” the highquality firm by deviating to a quality above q C .26 Since qLce = bL (q C , α) is a restricted best
response, firm L has no profitable deviation below q C . Firm L’s most profitable deviation
above q C is bH (q C , α). Our refinement on out-of-equilibrium beliefs implies that consumers
correctly anticipate that firm L deviated to bH (q C , α) when firm L unexpectedly obtains the
certificate; thus, this deviation yields ΠH (q C , bH (q C , α)). This deviation is unprofitable if and
only if condition (8) holds.
Similarly, condition (9) ensures that the high-quality producer has no incentive to deviate
from q C . The most profitable deviation is either to best-respond to qLce “from above,” that
is, by choosing bH (qLce , α), or to best-respond “from below,” by choosing bL (qLce , α). Our
refinement again implies that consumers correctly anticipate the most profitable among those
deviations, which yields max ΠH qLce , bH (qLce , α) , ΠL bL (qLce , α), qLce .
Define a certification threshold q C as feasible if it admits a certification equilibrium. De∗
note by QC the set of feasible certification thresholds. That is, QC is the set of all q C ≥ qH
satisfying conditions (8) and (9). For all q C ∈ QC , there exists an equilibrium in which
qH = q C and qL = bL (q C , α).
Lemma 3. The set of feasible certification thresholds QC is nonempty.
For simplicity, we will assume in what follows that the certification equilibrium is always
∗ , the certification equilibrium entails a
selected whenever it exists.27 Whenever q C > qH
higher quality for both firms than absent intervention. The high-quality firm produces a
higher quality in order to match the certification threshold. Anticipating this, the low-quality
firm also raises its quality because the quality levels are strategic complements. The fact
that firms underprovide quality in the absence of intervention (see Lemma 2) means that
certification can raise welfare.
As for the firms’ profits, the low-quality firm clearly benefits from less intense competition
resulting from more differentiated products. Moreover, it optimally augments the quality of
its product which further raises its profit.28 By contrast, the impact of certification on the
∗
Note that if q C = qH
, then condition (8) is more restrictive than the related “no-leapfrogging” condition in
the setup without certification. This is because uninformed consumers will revise their beliefs upon observing
that firm L unexpectedly obtains the certificate which makes deviating upwards more profitable now.
27
There is a second candidate equilibrium in which firms behave as if certification was not available and
∗
∗
install the quality levels (qL
, qH
) defined in Proposition 1. Uniqueness requires
∗
∗
∗
ΠH (qL
, qH
) < ΠH qL
, qC .
(10)
26
∗
∗
Condition (10) ensures that (qL
, qH
) cannot be an equilibrium since H could increase its profit by producing
C
goods of quality q and signal this deviation by means of the certificate.
R qce
28
ce
ce
∗
∗
∗
Formally, rewriting the change in firm L’s profit yields ΠL (qL
, qH
) − ΠL (qL
, qH
) = q∗H RqLH (qL
, q)dq +
H
R qce L
ce
L
ΠqL (q, qH )dq > 0 which is positive since the integrands of both terms are positive.
q∗
L
15
profit of the high-quality firm is ambiguous. On the one hand, certification helps the firm to
commit to producing a higher quality product, which – according to inequality (9) – increases
its profit when holding qL fixed at bL (q C , α). On the other hand, the low-quality firm reacts
to firm H’s increase to q C by adjusting its own quality upwards. This reduces the high-quality
firm’s profit. When the certification standard it set at a high level, so that a large investment
is necessary to obtain the certificate, firm H may thus earn less profit in the certification
equilibrium than without intervention.
The effect on consumers is a priori less clear-cut as they benefit from higher quality but
suffer from higher prices. Specifically, if consumers correctly anticipate the qualities qL and
R θ̂∗
R1
qH , then consumer surplus is CS(qL , qH ) ≡ x∗ qL (θ − x∗L ) dθ + θ̂∗ qH (θ − x∗H ) dθ. Taking
L
derivatives shows that CSqL > 0 is always true and that CSqH > 0 if and only if r > 45 .29
∗ ) > 0 which requires r ∗ > 7 .
The optimality condition (4) of firm L implies that RqLL (qL∗ , qH
4
Hence, certification increases consumer surplus in our setup.30
Having seen that welfare-enhancing certification is possible, we now turn to optimal certification. A welfare-maximizing regulator chooses q C to solve
max W (bL (q C , α), q C )
qC
subject to q C ∈ QC .
(11)
Among all feasible certification thresholds, the regulator chooses the one that maximizes
welfare, given that the high-quality firm exactly matches q C and the low-quality firm bestresponds to q C .
Our assumption that C 000 ≥ 0 is sufficient for concavity of W (bL (q C , α), q C ) in q C .31 Therefore, problem (11) has a unique unconstrained maximizer q T B ≡ arg maxq W (bL (q, α), q).32
Letting q̄ ≡ max QC , we can state the following result.
Proposition 2. If α is sufficiently small and ΠM (q T B ) > 0, then q T B ∈ QC , and the optimal
F I . If α is sufficiently large, then q T B ∈
certification standard is q T B > qH
/ QC , and the optimal
certification standard is q̄, which solves
ΠH bL (q̄, α), q̄ = ΠH bL (q̄, α), bH (bL (q̄, α), α) .
Proposition 2 contains two main results. When the share of informed consumers is small,
implementing q C = q T B is feasible and thus optimal. The condition ΠM (q T B ) > 0 is essentially an assumption on the cost function; it says that a monopolist must be able to profitably
r (8r 2 −6r−5)
r 2 (28r+5)
H r(4r+5)
Formally, CS = q2(4r−1)
.
2 , CSqL = 2(4r−1)3 and CSqH =
(4r−1)3
30
Note that although aggregate consumer surplus increases, the equilibrium utility of some consumers may
shrink.
31
See the proof of Proposition 2 for details.
32
The superscript TB (“third best”) indicates that in contrast to the second-best quality levels, only the
quality of firm H can be chosen by the regulator while firm L chooses a best response.
29
16
produce the third-best quality. When the share of informed consumers is large, implementing
q T B is no longer feasible. The regulator therefore chooses the largest feasible certification
standard, q̄.
F I , i.e., the third-best quality exceeds
To understand this result, note first that q T B > qH
the level that the high-quality firm would choose under full information. Because the secondbest qualities are above the unregulated full-information qualities, the third-best quality also
is. The second observation is that, for α = 1, the highest feasible certification standard is
F I . When all consumers observe quality, there is no reason for the high-quality firm to adopt
qH
a certificate to signal its quality. It will only adopt the certificate if the threshold is at the
level of quality it would have chosen anyway. Taken together, these two facts imply that q T B
is not feasible when α = 1, and by continuity also in its vicinity.
To see that q T B is feasible when α is small, recall that both firms’ best responses tend
to zero as the number of informed consumers approaches zero, i.e., bi (q, α) → 0 as α → 0
for i = H, L. Thus, the high-quality firm’s profit from foregoing the certificate also tends to
zero. By contrast, its profit from obtaining the certificate approaches the monopoly profit,
as the low-quality firm produces a good whose quality is close to zero. By assumption, the
monopoly profit of producing q T B is positive. Hence, for small α, q T B is in the set of feasible
certification thresholds QC .
When the share of informed consumers is low, the previous result implies that certification
can induce firm H to produce a quality exceeding the full-information level. The firm is willing
to install even expensive production technologies to obtain the certificate because its deviation
profit is small. Hence, if sufficiently few consumers observe quality, the regulator can exploit
the high-quality firm’s dependence on certification in order to raise the supplied quality of
firm H further towards the socially optimal level than it could by disclosing the exact quality
levels on offer.
Corollary. There exists a range of α over which the optimal certification standard decreases.
This corollary follows directly from the fact that the optimal certification standard is
F I for small α and q̄ for large α, noting that q̄ = q F I for α = 1. It suggests that as
q T B > qH
H
the share of informed consumers increases, the regulator may have to reduce the threshold
for certification.
5.2
Minimum quality standard
An MQS forces active firms to produce at least a quality of q M QS . Ronnen (1991) shows that
when all consumers are informed, introducing a suitable MQS is always welfare enhancing. If
both firms enter, an MQS that is set above qL∗ forces firm L to install a higher quality than
17
it otherwise would. In response, firm H also increases its quality. Ronnen (1991) shows that
under an MQS the goods are less differentiated than in the unregulated market, which leads
to lower quality-deflated prices and thus to higher participation.
The downside of adopting an MQS is that it may reduce the number of active firms. An
MQS introduces a lower bound on the investment in quality that is required to enter the
market, and thus plays a role that is similar to an entry cost. An MQS which is set too
restrictive does not allow both firms to recoup their initial investments in the production
technology. This may result in only one firm entering or no firm at all entering. In order to
maintain the competitive pressure on prices, the regulator has to set a rather low MQS so
as to ensure that both firms enter. In setting the level of q M QS , the regulator thus faces a
tradeoff between improving quality and maintaining price competition.
The negative impact of an MQS on entry is particularly severe if only few consumers
observe quality. As discussed in Section 4, the firms then cannot credibly produce high
qualities. The equilibrium level of vertical differentiation is low and firms earn little profit.
Adopting an MQS in such a situation further squeezes the firms’ profits. Even a moderate
MQS may deter one of them from entering, so only a relatively low MQS guarantees that
both firms enter. It follows that when the share of informed consumers is small the optimal
MQS may well be such that only one firm enters. As the following proposition shows, the
regulator may prefer a monopoly producing high quality to a duopoly producing low quality.
SB ≡ arg max W M (q). If α is sufficiently small and ΠM q SB > 0,
Proposition 3. Let qM
q
M
SB .
the optimal minimum quality standard is such that only one firm enters, and q M QS = qM
Proposition 3 extends the analysis in Ronnen (1991) to situations where the number of
informed consumers is small. For the reasons discussed above, when α → 0 the highest q M QS
that allows two firms to earn non-negative profits is close to zero. Having a single firm enter
the market, and controlling the firm’s quality through a more ambitious MQS, then yields
SB > 0, so that a monopolist can profitably produce the second-best
higher welfare. If ΠM qM
SB .
monopoly quality, it is optimal to set q M QS = qM
The following proposition uses these insights together with those in Ronnen (1991) to
assess under which conditions an MQS outperforms certification.
SB > 0. If α is sufficiently small, optimal certification leads
Proposition 4. Suppose ΠM qM
to higher welfare than an optimal MQS. If α is sufficiently large, an optimal MQS leads to
higher welfare than optimal certification.
To see the intuition behind Proposition 4, consider first the case where almost none of
the consumers observe quality. By Proposition 3, it is then optimal to set the MQS at
18
SB , so that only one firm enters. In this case, certification can do better, since
q M QS = qM
SB will generally induce two firms to enter, with firm H producing q = q SB .
setting q C = qM
H
M
Although firm L may credibly produce only at a low quality level, it exerts some competitive
pressure on firm H, which keeps the ensuing prices low and thus helps to improve welfare.
SB ,
Since any optimal certification standard q C must lead to weakly higher welfare than q C = qM
optimal certification performs better than an MQS.
If almost all consumers observe quality, we know from Proposition 2 that the certification
∗ . As α → 1, the
threshold has to be set close to the unregulated equilibrium quality level qH
welfare improvement that certification can bring about vanishes. By contrast, for α → 1, our
model converges to that of Ronnen (1991), who has shown that the optimal MQS improves
welfare even if all consumers are informed. We thus conclude that an MQS performs better
than certification when almost all consumers are informed.
6
Conclusion
This paper has investigated the effects of certification and minimum quality standards when
some consumers cannot discern the quality of traded goods. The presence of uninformed
consumers reduces firms’ incentives to invest in quality. In equilibrium, quality is undersupplied, which reduces both consumer surplus and profits. By certifying their products,
firms can demonstrate that their goods are of higher quality than consumers would otherwise expect. We have considered a simple form of government certification whereby firms are
awarded a certificate if they produce goods whose quality is above a publicly known threshold. The certification standard must be set low enough to raise firms’ profits when making
the investment needed to attain the threshold. Nevertheless, certification can induce some
firms to raise their quality above the highest level that would be attained if all consumers
were informed. When the share of uninformed consumers is large, firms are more reliant on
certification. This allows the government to implement an ambitious certification threshold.
We have also compared certification to an MQS. When the share of informed consumers
is small, certification may be preferred over an MQS, since the latter potentially deters firms
from entering the market. By contrast, when the share of informed consumers is high, an
MQS tends to be more effective than certification.
Our results could be extended in several directions by further research. So far, we have
analyzed the relative merits of certification and MQS, but have not considered adopting
both instruments together. Introducing both instruments would improve the government’s
ability to manipulate the quality of active firms. In particular, if only few consumers are
informed and two firms have entered, then a suitable certification standard allows firm H
19
to provide high quality goods and the increased differentiation results in higher revenues.
Therefore, complementing an MQS with certification may alleviate the entry-deterring effect
of a minimum quality standard. Certification may thus be particularly valuable when used
in conjunction with an MQS.
The industry’s ability to provide a certification system itself is also an interesting point
that deserves further attention. We have pointed out that certification standards may increase
the profits of all active firms. This suggests that firms may have an interest in building up their
own certifying institution whose certification scheme is designed to maximize the industry
profit rather than welfare. Why do we observe that firms sometimes rely on government
certification? One important issue from which we have abstracted in our model is that
a certifier must have correct incentives for designing and enforcing a certification scheme
honestly.33 The recent experiences with private certifiers that have issued inflated ratings for
financial products suggest that this requirement is more likely to be satisfied by government
certification.34 Nevertheless, the availability of privately run certification may restrict the
government’s leeway to manipulate the firms’ qualities by means of certification.
Appendix A
Proofs
Proof of Proposition 1
∗ ) satisfying the necessary conditions (4)
We first show that there is a unique solution (qL∗ , qH
∗ ) and that q ∗ and q ∗
and (6). We then show that both firms earn positive payoffs at (qL∗ , qH
H
L
are global best responses when C 000 ≥ 0. Together these properties establish existence and
uniqueness of equilibrium. Finally, we show that the equilibrium qualities increase with α.
Let q max > 0 denote the unique solution to the equation αRqHH (q, q) − C 0 (q) = 0. Define
B(q, α) ≡ bL bH (q, α), α −q on [0, q max ]. By the properties of bL and bH , B is continuous and
L
L
strictly decreasing in q as Bq (q) = bH
qL bqH − 1 < 0. To see this, note that RqL is homogeneous
of degree zero (i.e., qL RqLL ,qL + qH RqLL ,qH = 0), implying
bL
qH (qH , α) = −
RqLL ,qH
RqLL ,qH
qL
<
−
=
.
L
00
L
RqL ,qL − C /α
RqL ,qL
qH
Homogeneity of degree zero of RqHH similarly implies
bH
qL (qL , α) = −
RqHH ,qL
RqHH ,qL
qH
<
−
=
.
H
00
H
RqH ,qH − C /α
RqH ,qH
qL
L
max , α) < 0.
It follows that bH
qL bqH < r(1/r) = 1. Moreover, we have B(0, α) > 0 and B(q
33
34
Gehrig and Jost (1995) studies the incentives of self-regulating organizations to conduct costly monitoring.
See e.g. Mathis, McAndrews, and Rochet (2009).
20
∗ = bH (q ∗ , α)
Thus, there exists a unique qL∗ such that B(qL∗ ) = 0. By construction, qL∗ and qH
L
satisfy (4) and (6).
Because qL is bounded below by zero, RqLL (0, qH ) = 1/16 > 0 for all qH > 0, and
C 0 (0) = 0, the low-quality firm can always guarantee itself a positive payoff when entering,
∗ ) > 0.
so ΠL (qL∗ , qH
∗ ) > ΠL (q ∗ , q ∗ ), implying that the high-quality firm earns
We now show that ΠH (qL∗ , qH
L H
a positive payoff as well. From (1) and (2), we obtain RH (qL , qH ) − RL (qL , qH ) = (qH −
qL )r/(4r − 1). Using the explicit expression for RqHH given in Footnote 13, we have
4r(2 − 3r + 4r2 )
r
⇔ 4r − 7 > 0.
> RqHH (qL , qH ) =
4r − 1
(4r − 1)3
In equilibrium, this inequality must hold because RqLL > 0 requires r∗ > 7/4, implying
∗ ) − RL (q ∗ , q ∗ ) ≥ (q ∗ − q ∗ ) RH (q ∗ , q ∗ ). We thus have
RH (qL∗ , qH
qH L H
L H
H
L
∗
∗
∗
∗
α RH (qL∗ , qH
) − RL (qL∗ , qH
) > α(qH
− qL∗ )RqHH (qL∗ , qH
)
=
∗
(qH
−
∗
qL∗ )C 0 (qH
)
Z
∗
qH
>
∗
qL
∗
C 0 (q)dq = C(qH
) − C(qL∗ ),
where the equality follows from the first-order condition (6) and the last inequality from the
∗ ) − RL (q ∗ , q ∗ ) > 0 and ΠH (q ∗ , q ∗ ) > ΠL (q ∗ , q ∗ ).
convexity of C. Hence, RH (qL∗ , qH
L H
L H
L H
∗ are global best responses when uninformed consumers’
We now check that qL∗ and qH
∗ . We begin with q ∗ . Clearly, deviating from q ∗ does not
beliefs are q̂L = qL∗ and q̂H = qH
H
H
affect second-stage revenues, so it suffices to show that
∗
∗
αRH (qL∗ , qH
) − C(qH
) ≥ max∗ αRL (q, qL∗ ) − C(q) .
q≤qL
We have
∗
∗
∗
αRH (qL∗ , qH
) − C(qH
) > αRL (qL∗ , qH
) − C(qL∗ )
∗
= max
αRL (q, qH
) − C(q) > max∗ αRL (q, qL∗ ) − C(q),
∗
q≤qH
q≤qL
where the first inequality follows from our previous result and the second from the envelope
theorem, noting that
∂
L
max αR (q, qH ) − C(q) = αRqLH > 0.
∂qH q≤qH
Turning to qL∗ , similarly as above we need to show that
∗
∗
αRL (qL∗ , qH
) − C(qL∗ ) ≥ max
αRH (qH
, q) − C(q).
∗
q≥qH
21
RqHH , C 0
C 0 (q̃)
C 0 (q) + (q̃ − q)C 00 (q)
4α
9
αRqHH (q, q)
αRqHH (q, q̃)
q
δ
bH (q, α) q + ∆
q̃
Figure 3: Proof of Proposition 1
∗ , q) − C(q) < 0 for all q ≥ q ∗ , which,
We now establish that C 000 ≥ 0 implies that αRH (qH
H
∗ ) − C(q ∗ ) > 0, suffices for the required result. Using
together with the fact that αRL (qL∗ , qH
L
RH (q, q) = 0 and concavity in own quality (RqHH ,qH < 0), we can write
αR
H
H
q, b (q, α) − C bH (q, α) =
bH (q,α) Z
q
αRqHH (q, q̃) − C 0 (q̃) dq̃ − C(q).
The following arguments are illustrated in Figure 3. First, we have
Z bH (q,α)
Z q+∆
H
H
0
αRqH (q, q̃) − C (q̃) dq̃ <
αRqH − C 0 (q) + (q̃ − q)C 00 (q) dq̃
q
q
=
2
αRqHH (q, q) − C 0 (q)
,
2C 00 (q)
where ∆ ≡ αRqHH (q, q) − C 0 (q) /C 00 (q) and the inequality is due to the concavity of RH in
qH and C 000 ≥ 0. Second, we have
Z q
Z
0
C(q) =
C (q̃)dq̃ ≥
0
δ
q
0
(C 0 (q))2
C (q) − (q − q̃)C 00 (q) dq̃ =
,
2C 00 (q)
where δ ≡ q − C 0 (q)/C 00 (q) and the inequality is due to C 000 ≥ 0.
Putting both together yields
αRqHH (q, q)
H
0
αRH q, bH (q, α) − C bH (q, α) <
αR
(q,
q)
−
2C
(q)
.
q
H
2C 00 (q)
Clearly, αRH q, bH (q, α) − C bH (q, α) < 0 if 2C 0 (q) ≥ αRqHH (q, q). Condition (6) implies
0
∗ ) = αRH (q ∗ , q ∗ ) > α/4 because RH is bounded below by 1 . Hence, 2C (q ∗ ) >
that C 0 (qH
qH L H
qH
H
4
H
H
∗
H
∗
H
∗
α/2 > 4α/9 = αRqH (q, q), implying that αR qH , b (qH , α) − C b (qH , α) < 0, so
∗ are unprofitable.
deviations to q ≥ qH
22
∗ increase with α. Recall that q ∗ necessarily satisfies
Finally, we establish that qL∗ and qH
L
H
∗
L
B(qL , α) = 0. We have Bq (q, α) < 0 on the relevant range, and Bα = bL
α + bqH bα > 0,
where the arguments are omitted for brevity. By the implicit function theorem, qL∗ thus
∗ = bH (q ∗ , α) and bH > 0, q ∗ also increases in α.
increases in α. Since qH
qL
L
H
Proof of Lemma 2
∗ ) ≥ 0, together implying that W (q ∗ , q ∗ ) >
We first show that Wqi > Πiqi and that Πiqi (qL∗ , qH
qi L H
qH (12r2 −r−2)
denote welfare gross
0, i = L, H. Let U (qL , qH ) ≡ W (qL , qH ) + C(qL ) + C(qH ) = 2(4r−1)2
of investment costs. We have WqL (qL , qH ) = UqL (qL , qH ) − C 0 (qL ) > RqLL (qL , qH ) − C 0 (qL ) =
ΠL
qL (qL , qH ) if and only if
UqL (qL , qH ) > RqLL (qL , qH )
⇐⇒
r2 (4r − 7)
r2 (20r − 17)
>
,
2(4r − 1)3
(4r − 1)3
which is always true for r > 1 because 20r − 17 > 2 (4r − 7) ⇔ r ≥ 1/4. We also have
WqH (qL , qH ) > ΠH
qL (qL , qH ) if and only if
UqH (qL , qH ) > RqHH (qL , qH )
⇐⇒
24r3 − 18r2 + 5r + 1
4r(2 − 3r + 4r2 )
>
,
(4r − 1)3
(4r − 1)3
which holds because 24r3 − 18r2 + 5r + 1 > 4r(2 − 3r + 4r2 ) ⇔ 8r3 − 6r2 − 3r + 1 =
(4r − 1)(2r2 − r − 1) > 0 is true for any r > 1. From the equilibrium conditions (4) and (6) we
∗ ) = (1−α)Ri (q ∗ , q ∗ ) ≥ 0 and therefore W (q ∗ , q ∗ ) > (1−α)Ri (q ∗ , q ∗ ) ≥ 0.
have Πiqi (qL∗ , qH
qi L H
qi L H
qi L H
Next, we establish concavity of U . Differentiating UqL and UqH yields
r3 (4r + 17)
<0
qH (4r − 1)4
r(4r + 17)
=−
<0
qH (4r − 1)4
r2 (4r + 17)
=
= WqL ,qH > 0,
qH (4r − 1)4
UqL ,qL = −
UqH ,qH
UqL ,qH
from which we obtain the determinant of the Hessian of U ,
UqL ,qL UqH ,qH − (UqL ,qH )2 = 0.
Hence, the Hessian of U is negative semi-definite, implying that U is concave. Moreover, since
∗ ) > 0 with strict concavity
C is strictly convex, W is strictly concave. Combining Wqi (qL∗ , qH
of W and WqL ,qH > 0 yields the claimed result.
Proof of Lemma 3
FI.
To establish that QC is nonempty, we first prove that condition (8) holds for all q C ≥ qM
∗ such that condition (9) is satisfied for all q C ∈ [q ∗ , q † ].
Then, we show that there exists q † ≥ qH
H
F I , q ∗ } ≤ q † , with strict inequality for α < 1.
Finally, we show that max{qM
H
23
We know from the proof of Proposition 1 that C 000 ≥ 0 implies αRH q, bH (q, α) −
C bH (q, α) < 0 if 2C 0 (q) ≥ αRqHH (q, q) = 4α/9. Applying this result for α = 1, a sufficient
condition for deviations from qLce to bH (q C , 1) to be unprofitable is 2C 0 (q C ) ≥ 4/9. We have
F I ) = RM q F I = 1/4 and thus 2C 0 (q F I ) = 1/2 > 4/9. Because C 0 is increasing, the
C 0 (qM
qM
M
M
FI.
condition is satisfied for all q C ≥ qM
Define q † as the solution to q † = bH bL (q † , α), 1 . A similar reasoning as in the proof of
Proposition 1 implies existence and uniqueness of q † . Condition (9) has two parts. Consider
∗ , we have q ∗ ≤
first the inequality ΠH (qLce , q C ) ≥ ΠH (qLce , bH (qLce , α)). For any q C ≥ qH
H
bH (qLce , α) ≤ q C . Thus,
ΠH (qLce , q C ) − ΠH (qLce , bH (qLce , α)) =
Z
qC
ce ,α)
bH (qL
[RqHH (qLce , q) − C 0 (q)]dq.
We have RqHH (qLce , q) − C 0 (q) > 0 for q ∈ [bH (qLce , α), bH (qLce , 1)). From the definition of q † , it
∗ , q † ].
follows that ΠH (qLce , q C ) − ΠH (qLce , bH (qLce , α)) ≥ 0 for all q C ∈ [qH
∗ , q † ], there
Now consider the inequality ΠH (qLce , q C ) ≥ ΠL (bL (qLce , α), qLce ). For q C ∈ [qH
exists α̃ ≤ 1 such that bH (qLce , α̃) = q C . This follows from bH (qLce , α) ≤ q C ≤ bH (qLce , 1)
and continuity of bH in α. In Proposition 1 we have shown that if q C /qLce ≥ 7/4 (which is
necessarily satisfied since qLce = bL (q C , α)), then α̃RH (qLce , q C ) − C q C > α̃RL (qLce , q C ) −
C (qLce ), which implies RH (qLce , q C ) − C q C > RL (qLce , q C ) − C (qLce ) and thus ΠH (qLce , q C ) >
ΠL (qLce , q C ). It remains to be shown that ΠL (qLce , q C ) ≥ ΠL (bL (qLce , α), qLce ). We have
ΠL (qLce , q C ) − ΠL (bL (qLce , α), qLce ) = ΠL (qLce , q C ) − ΠL (bL (qLce , α), q C )
+ ΠL (bL (qLce , α), q C ) − ΠL (bL (qLce , α), qLce )
Z qce
Z qC
L
C
=
ΠL
(q,
q
)dq
+
RqLH (bL (qLce , α), q)dq > 0,
qL
ce ,α)
bL (qL
ce
qL
∗ , q † ],
where the inequality follows from both integrands being positive. Hence, for q C ∈ [qH
condition (9) is satisfied.
∗ follows from bH being increasing in q and α as well as bL being increasing
Finally, q † ≥ qH
L
F I follows from q F I = bH (0, 1), q ce > 0, and bH being increasing in q .
in qH , while q † ≥ qM
L
M
L
Proof of Proposition 2
We start by showing that q T B ∈ QC for small α. Because limα→0 bL (q, α) = 0 and limα→0 bH (q, α) =
q for any q ≥ 0, we have
lim ΠH bL (q, α), bH bL (q, α), α
α→0
= ΠL bL bL (q, α), α , bL (q, α) = 0
for any q ≥ 0. By assumption, ΠM (q T B ) = ΠH (0, q T B ) > 0. Therefore, there exists α̃ > 0
such that conditions (8) and (9) are satisfied for all α ≤ α̃ when q C = q T B , implying q T B ∈ QC .
24
Since q T B is the unconstrained maximizer of W (bL (q C , α), q C ), it is the solution to (11)
whenever α ≤ α̃.
F I . Note first that W ((bL (q C , α), q C ) is concave in q C when
We now show that q T B > qH
C 000 ≥ 0. In fact, we have
d2 W (bL (q, α), q)
L
= WqL ,qL bL
qH + WqL bqH ,qH + WqH ,qH .
dq 2
The proof of Lemma 2 establishes that WqL ,qL < 0 and WqH ,qH < 0. Moreover, WqL (bL (q C ), q C ) >
L
0 over the relevant range of q C , and bL
qH > 0, so what remains to be shown is that bqH ,qH ≤ 0.
We have
bL
qH ,qH
=
L
RqLL ,qL ,qH bL
qH + RqL ,qH ,qH
000
L
C 00 /α − RqLL ,qL − RqLL ,qH bL
qH C /α − RqL ,qL ,qL
.
2
C 00 /α − RqLL ,qL
Computations yield
RqLL ,qL ,qL = −
RqLL ,qL ,qH =
6r4 (20r + 7)
2 (4r − 1)5 < 0
qH
4r3 (16r2 + 40r + 7)
>0
2 (4r − 1)5
qH
RqLL ,qH ,qH = −
2r2 (64r2 + 100r + 7)
< 0.
2 (4r − 1)5
qH
L
L
L
Because C 000 ≥ 0 by assumption, it follows that bL
qH ,qH ≤ 0 if RqL ,qL ,qH bqH + RqL ,qH ,qH ≤ 0.
Using the fact that, as shown in the proof of Lemma 1, bL
qH < qL /qH , we obtain
RqLL ,qL ,qH
+ RqLL ,qH ,qH
r
4r2 (16r2 + 40r + 7) − 2r2 (64r2 + 100r + 7)
=
≤0
2 (4r − 1)5
qH
L
RqLL ,qL ,qH bL
qH + RqL ,qH ,qH <
⇐⇒ − 32r2 − 20r + 7 ≤ 0,
which is always satisfied since r > 1. Concavity in q C implies that a sufficient condition for
F I is
q T B > qH
dW (bL (q C , α), q C ) H
FI FI
0 FI
C F I > RqH (qL , qH ) − C (qH ) = 0.
dq C
q =q
(12)
H
We have
dW (bL (q C , α), q C ) L
FI
FI FI
FI FI
C F I = bqH (qH , α)WqL (qL , qH ) + WqH (qL , qH ).
dq C
q =q
H
F I ) > 0, a sufficient condition for (12) is
As we know from the proof of Lemma 2, WqL (qLF I , qH
FI
FI
FI
WqH (qLF I , qH
) + C 0 (qH
) > RqHH (qLF I , qH
).
25
We have WqH (qL , qH )+C 0 (qH ) =
24r3 −18r2 +5r+1
,
(4r−1)3
which is bounded below by 3/8. In any equi-
FI >
librium with α = 1, the first-order condition of the low-quality firm implies r > 7/4, so qH
4r(2−3r+4r2 )
F I ) < RH (q , (7/4)q ) =
is decreasing in r, RqHH (qLF I , qH
L
qH L
(4r−1)3
F
I
F
I
0
F
I
7/24 < 3/8 < WqH (qL , qH ) + C (qH ). Hence, (12) must hold.
Next, we establish that q T B ∈
/ QC for large α. Define the highest feasible certification
(7/4)qLF I . Because RqHH =
standard as q̄(α) ≡ max{q|∆Π(q, α) ≥ 0}, where
∆Π(q, α) ≡ ΠH bL (q, α), q − ΠH bL (q, α), bH (bL (q, α), α) .
Because ΠH is continuous and bL and bH are continuous, ∆Π is continuous in q and α. By
F I , implying
the properties of bH (single-valued best response), ∆Π(q, 1) < 0 for all q > qH
F I . Moreover, lim
q̄(1) = qH
q→∞ ∆Π(q, 1) = −∞.
F I = q̄(1). Thus, q T B ∈
We have shown previously that q T B > qH
/ QC for α = 1. What
remains to be shown is that this also holds in the vicinity of α = 1. Because of pointwise
continuity of ∆Π in α, for any q, limα→1 ∆Π(q, α) = ∆Π(q, 1). For any δ > 0, there is hence
F I + δ. Since ∂∆Π/∂α is bounded above,
some > 0, such that ∆Π(q, 1) ≤ − for all q ≥ qH
F I + δ. Therefore, q̄(α) approaches or lies below q F I as
for limα→1− ∆Π(q, 1) < 0 for all q ≥ qH
H
F I < qT B , qT B ∈
α → 1. Since qH
/ QC as α → 1.
Proof of Proposition 3
0
Define q max (α) as the unique solution to αRqHH (q, q) − C (q) = 0, giving the upper bound
of the domain on which bH (q, α) is defined, with bH (q max (α), α) = q max (α). Note that
ΠH q max (α), bH (q max (α), α) = ΠH (q max (α), q max (α)) = −C(q max (α)) < 0. It follows from
RqHH (q, q) = 4/9 that limα→0 q max (α) = 0.
Define the highest MQS that entails an equilibrium in which two firms enter by q̂(α) ≡
max q ≥ qL∗ |ΠL q, bH (q, α) ≥ 0 ∧ ΠH q, bH (q, α) ≥ 0 . Since ΠL and ΠH are continuous,
∗ ) > 0 and ΠH (q ∗ , q ∗ ) > 0 as well as ΠH q max (α), bH (q max (α), α) < 0,
and since ΠL (qL∗ , qH
L H
we know that for α > 0, q̂(α) exists, is unique, and satisfies qL∗ (α) < q̂(α) < q max (α).
Combining limα→0 q max (α) = 0 and q̂(α) ≥ 0 yields limα→0 q̂(α) = 0.
Next, we show that if α is small enough, then it is optimal to set the MQS such that
SB ) > max
M QS , bH (q M QS , α)). Let r M QS ≡
only one firm enters, i.e., W M (qM
q M QS ≤q̂(α) W (q
(12r2 −r−2)
bH (q M QS )
and ψ(r) ≡ 2(4r−1)2 so that W (qL , qH ) = qH ψ (qH /qL ). Note that maxr ψ(r) =
q M QS
ψ(1) = 1/2. We now derive an upper bound on the welfare from an MQS that induces both
26
firms to enter:
max
q M QS ≤q̂(α)
W (q M QS , bH (q M QS , α)) =
max
q M QS ≤q̂(α)
bH (q M QS , α)ψ(rM QS )
− C(q M QS ) − C(bH (q M QS , α))
≤
=
max
q M QS ≤q̂(α)
max
bH (q M QS , α) max ψ(r)
r
bH (q M QS , α)
q M QS ≤q̂(α)
2
≤
q max (α)
,
2
where the last inequality holds because bH (q, α) ≤ q max (α). Together with the fact that
q max (α) → 0 as α → 0, this implies
lim
max
α→0 q M QS ≤q̂(α)
W (q M QS , bH (q M QS )) = 0.
SB ) > 0 with q SB being characterized
If only one firm enters, the optimal welfare is W M (qM
M
SB = C 0 (q SB ). Thus, W M (q SB ) > max
M QS , bH (q M QS )) for α sufficiently
by 83 qM
q M QS ≤q̂(α) W (q
M
M
SB ) > 0, so q M QS∗ = q SB makes entry by one firm profitable.
small. By assumption, ΠM (qM
M
Proof of Proposition 4
SB . Clearly,
We first prove the result for α → 0. Consider the certification level q C = qM
SB > ΠL q, q SB ≥
SB = W M (q SB ) since W
SB , α), q SB > lim
W bL (qM
qL q, qM
qL →0 W qL , qM
qL
M
M
M
SB , α) .
0 for q ∈ 0, bL (qM
SB is feasible, note that firm H’s profit when producing q C as α → 0
To see that q C = qM
SB , α), q SB = ΠM (q SB ), which is strictly positive by assumption. Its
is limα→0 ΠH bL (qM
M
M
deviation payoffs are
SB
SB
lim ΠH bL (qM
, α), bH bL (qM
, α), α = 0
α→0
SB
SB
lim ΠL bL bL (qM
, α), α , bL (qM
, α) = 0.
α→0
SB , α), q SB = 0,
Firm L’s payoff when best-responding to q C from below is limα→0 ΠL bL (qM
M
SB , bH q SB , α
SB , q SB ) = −C(q SB ) < 0.
while its deviation payoff is limα→0 ΠH qM
= ΠH (qM
M
M
M
Therefore, conditions (8) and (9) are satisfied for α sufficiently small. Finally, we have
SB , α), q SB > W M q SB , establishing the claimed
maxqC ∈QC W bL q C , α , q C ≥ W bL (qM
M
M
result that certification outperforms an MQS for small α.
The argument for our result that an MQS outperforms certification as α → 1 is that,
when all consumers observe quality, certification becomes pointless and cannot improve on
the unregulated outcome. Moreover, for α → 1, our model converges to Ronnen (1991), who
shows that there exists a welfare-increasing q M QS > 0.
27
Appendix B
Private and social incentives to increase quality
To understand Lemma 2, it is useful to decompose the private and social incentive to increase
quality into several parts. We do this here for the high-quality firm; the analysis for the
low-quality firm is similar. Rewrite W as
Z
Z 1
θdθ + (qH − qL )
W (qL , qH ) = qL
x∗L
Applying Leibniz’ rule and noting that
1
θ̂∗
R1
θ̂∗
θdθ − C(qL ) − C(qH ).
2
θdθ = 1/2 − θ̂∗ /2 = (1 − θ̂∗ )(1 + θ̂∗ )/2, the
marginal welfare effect of an increase in qH can be written as
WqH = (1 − θ̂∗ )
∂x∗L
1 + θ̂∗
∂ θ̂∗
−
(qH − qL )θ̂∗ −
qL x∗L − C 0 (qH ).
2
∂qH
∂qH
(13)
The first term represents the average high-quality consumer’s taste for quality (weighted
by the share of consumers buying high quality). The second and third term correspond to
the negative effects on welfare that work through the equilibrium prices: an increase in qH
leads to higher prices, leading some consumers to shift from high to low quality and others
to leave the market. Formally, we have ∂ θ̂∗ /∂qH = 2qL /(4qH − qL )2 > 0 and ∂x∗L /∂qH =
3qL /(4qH − qL )2 > 0. The fourth term represents the cost effect.
Now consider the private incentive to raise quality. Writing firm H’s revenue per consumer
as RH (qL , qH ) = maxpH pH 1 − θ̂(qL , qH , pL , pH ) and applying the envelope theorem, we
obtain, at pL = p∗L = qL x∗L ,
RqHH
= −pH
∂ θ̂
∂ θ̂ ∂p∗L
+
∂qH
∂pL ∂qH
!
.
(14)
From the definition of θ̂, we have ∂ θ̂/∂qH = −θ̂/(qH − qL ), ∂ θ̂/∂pH = 1/(qH − qL ), and
∂ θ̂/∂pL = −1/(qH − qL ). The first-order condition of firm H’s pricing problem gives us pH =
1−θ̂
∂ θ̂/∂pH
= (1− θ̂)(qH −qL ). Replacing these expressions in (14), using ∂p∗L /∂qH = qL ∂x∗L /∂qH ,
H
0
and noting that ΠH
qH = RqH − C (qH ) yields (15). Thus, when all consumers are informed,
∂x∗L
H
∗
∗
ΠqH = (1 − θ̂ ) θ̂ + qL
− C 0 (qH ).
(15)
∂qH
The term in square brackets comprises the marginal consumer’s taste for quality, θ̂∗ , as well
as a competition-relaxing effect of higher quality, which works through the low-quality firm’s
price, x∗L . The second term again represents the cost effect.
Subtracting (15) from (13) yields
"
#
∂x∗L
1 + θ̂∗
dθ̂∗
H
∗
∗
WqH − ΠqH = (1 − θ̂ )
− θ̂ −
(qH − qL )θ̂∗ −
qL (x∗L + 1 − θ̂∗ ).
2
dqH
∂qH
28
(16)
Expression (16) shows that there are several reasons for differences between the social and
the private incentive to increase quality. On the one hand, firms care about the marginal
consumer whereas the social planner cares about the average consumer, as expressed in the
first bracket. Because the average consumer of high quality has a stronger taste for quality
than the marginal consumer ((1 + θ̂∗ )/2 > θ̂∗ ), this tends to give the planner a stronger
incentive to increase quality than the firm. On the other hand, an increase in quality leads to
higher equilibrium prices. This is detrimental to welfare but beneficial to the firm, and thus
tends to give the firm a stronger incentive to raise quality than the planner.
References
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